Method for correcting low permeability laboratory measurements for leaks

ABSTRACT

A method for correcting low permeability laboratory measurements for leaks. A pulse-decay permeability (PDP) experiment is performed on a core sample retrieved from a formation. The PDP experiment includes flowing fluid through the core sample in a sealed enclosure. In response to flowing the fluid through the core sample, a change in fluid pressure is measured over time. Based on the change in fluid pressure over time, a leakage of fluid from the sealed enclosure is determined. In response to determining the leakage of fluid from the sealed enclosure, an analytical model of the leakage is determined based on the change in fluid pressure over time.

CROSS-REFERENCE TO RELATED APPLICATION(S)

This application claims the benefit of priority to U.S. ProvisionalPatent Application No. 62/521,786, filed Jun. 19, 2017 and entitled“METHOD FOR CORRECTING LOW PERMEABILITY LABORATORY MEASUREMENTS FORLEAKS”, the contents of which are hereby incorporated by reference.

TECHNICAL FIELD

This specification relates to measuring permeability of low permeabilityrock formations.

BACKGROUND

Tight rock formations such as shales have attracted increased attentionas potential hydrocarbon resources due to the development of technologylike hydraulic fracturing and horizontal drilling, which has allowedproduction from these unconventional resources to become economicallyviable. The permeability of such rock formations is a parameter tocharacterize the reservoir, and is used to predict a reservoir'sproductivity and profitability. Accurately measuring the permeability ofsuch unconventional resources is useful in determining the ability toproduce from a reservoir.

SUMMARY

This specification describes technologies relating to correcting lowpermeability laboratory measurements for leaks.

Certain aspects of the subject matter described here can be implementedas a method. A pulse-decay permeability (PDP) experiment is performed ona core sample retrieved from a formation. The PDP experiment includesflowing fluid through the core sample in a sealed enclosure. In responseto flowing the fluid through the core sample, a change in fluid pressureis measured over time. Based on the change in fluid pressure over time,a leakage of fluid from the sealed enclosure is determined. In responseto determining the leakage of fluid from the sealed enclosure, ananalytical model of the leakage is determined based on the change influid pressure over time.

This, and other aspects, can include one or more of the followingfeatures.

Based on the change in fluid pressure over time and on the analyticalmodel of the leakage, a permeability model representing a permeabilityof the core sample can be determined.

The permeability can be determined by fitting the non-straight curvewith consideration of the leakage effect.

The sealed enclosure can include an upstream reservoir, a downstreamreservoir, and a core holder between the upstream reservoir and thedownstream reservoir. Performing the PDP experiment on the core samplecan include positioning the core sample in the core holder, flowing thefluid into the upstream reservoir, flowing the fluid through the coresample in the core holder, and flowing the fluid into the downstreamreservoir. The leakage of fluid from the upstream and downstreamreservoirs can be determined based on the pressure difference between anupstream reservoir and a downstream reservoir.

The core sample can be an unfractured core sample.

In response to flowing the fluid through the unfractured core sample,measuring the change in fluid pressure over time can include recordingpressure transient curves for each of the upstream and downstreamreservoirs.

A log of an experimental pressure difference between the upstream anddownstream reservoirs can be determined based on the pressure transientcurves.

Determining the leakage of the fluid based on the change in fluidpressure over time can include determining that the log of theexperimental pressure difference is substantially a straight line anddetermining an absence of the leakage from the upstream and downstreamreservoirs.

Determining the leakage of the fluid based on the change in fluidpressure over time can include determining that the log of theexperimental pressure difference is substantially a non-straight curveand determining a presence of leakage from the sealed enclosure.

In response to determining the leakage of fluid from the upstream anddownstream reservoirs, the leakage rate can be determined based on thechange in fluid pressure over time by determining a theoretical pressuredifference between the upstream and downstream reservoirs based onparameters of the fluid flowed through the unfractured core sample,where the theoretical pressure difference can be independent of theleakage from the sealed enclosure. In response to determining theleakage of fluid from the sealed enclosure, the leakage rate can bedetermined based on the change in fluid pressure over time by comparingthe theoretical pressure difference and the experimental pressuredifference.

The core sample can be a fractured core sample including a fractureformed in a matrix of the core sample.

In response to flowing the fluid through the fractured core sample,measuring the change in fluid pressure over time can include measuring afirst-stage change in fluid pressure over time, where the first-stagechange in fluid pressure can be based on flow of the fluid through thefracture. In response to flowing the fluid through the core sample,measuring the change in fluid pressure over time can include measuring asecond-stage change in fluid pressure over time, where the second-stagechange in fluid pressure can be based on flow of the fluid through thematrix after the flow of the fluid through the fracture. Based on thechange in fluid pressure over time, determining the leakage of fluidfrom the sealed enclosure can include determining that a log of thesecond-stage change in fluid pressure over time deviates from asubstantially straight line.

The details of one or more implementations of the subject matterdescribed in this specification are set forth in the accompanyingdrawings and the description later. Other features, aspects, andadvantages of the subject matter will become apparent from thedescription, the drawings, and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of an example core sample prepared for apulse-decay permeability (PDP) experiment.

FIGS. 2A, 2B, 2C, and 2D are simplified diagrams and pressure graphs ofexample PDP experiments.

FIGS. 3A and 3B are simplified flow graphs of example 1st-stage PDPexperiments.

FIGS. 4A and 4B are simplified flow graphs of example 2nd-stage PDPexperiments.

FIG. 5 is a flow chart illustrating an example method for a PDPexperiment.

FIGS. 6A, 6B, 6C, and 6D are various data plots of an example PDPexperiment.

FIGS. 7A, 7B, 7C, and 7D are various data plots of an example PDPexperiment.

Like reference numbers and designations in the various drawings indicatelike elements.

DETAILED DESCRIPTION

This specification describes a method for correcting low and ultralowtransient permeability measurements for instrument leaks during a corepermeability experiment utilizing gas as the fluid medium (for example,helium or nitrogen). The method is applicable to single and dualpermeability systems, corresponding to one- and two-stage pulse-decaypermeability (PDP) experiments, respectively. The first derivative (thatis, slope) of the log of the pressure difference (between the upstreamand downstream reservoirs) curve is calculated and plotted versus time.If this derivative line diverges from a constant-derivative line withtime, there exists gas leak in the measurement system. Based on thisdivergence, the leakage rate can be simultaneously calculated andcompensated for, in order to estimate the permeability of the samplemore accurately.

The pulse-decay permeability (PDP) method takes relatively less time tomeasure permeability in comparison to steady-state permeability methods,especially for tight, low permeability rock formations. The PDP methodpreserves the structure of the rock sample in contrast to crushed sampleexperiments, and it can also reflect anisotropic properties (forexample, horizontal and vertical permeability) of a rock formation byusing core samples with different orientations. The method described inthis specification can be implemented, so as to realize one or more ofthe following advantages. Permeability measurements can be corrected forleakage, if any leakage is present in the testing apparatus, therebyavoiding a potential need to rerun experiments. Because the structure ofthe rock sample is preserved, the method can be applicable to single anddual permeability systems, accurately measuring permeability bycorrecting for any potential leakage at each stage of the experiment.

FIG. 1 shows a schematic diagram for an example PDP experiment. Thesystem 100 includes an upstream reservoir 101, a downstream reservoir103, and a core holder 105. A core sample 107 is placed in the coreholder 105, which can hold a high hydrostatic confining pressure-usuallyaround 2,000 to 5,000 pounds per square inch gauge (psig)—and then atest gas (for example, helium or nitrogen) can be pumped into upstreamreservoir 101 and downstream reservoir 105 for enough time to ensure theupstream pressure, downstream pressure, and pore pressure reachequilibrium-usually around 1,000 to 2,000 psig. The high pressure canlessen the Klinkenberg slipping effect during gas transport.

The upstream reservoir 101 is provided a pulse pressure that is a smallpercentage of the initial gas pressure (for example, an additional 4% ofinitial 1,000 psi, or 40 psi) by pressure source 109. Due to the pulsepressure, the test gas flows from the upstream reservoir 101 to thedownstream reservoir 103 by passing through the core sample 107. Thepressures in the upstream reservoir 101 and downstream reservoir 103 arerecorded as a function of time, and the permeability of the core sample107 is estimated by the changes of pressures in the reservoirs.

Unfractured core samples can be described as single permeabilitysystems, which are characterized by matrix permeability. FIG. 2A shows asingle permeability system of matrix 201A, and FIG. 2B shows thecorresponding pressure transient curves for the upstream reservoir 101(curve 203) and downstream reservoir 103 (curve 205). For the graphshown in FIG. 2B, the x-axis is time in seconds (s), and the y-axis ispressure in pounds per square inch (psi). Due to the gas flow throughthe core sample 207A, the pressure of the upstream reservoir 101declines while the pressure of the downstream reservoir 103 increasesuntil the reservoirs reach equilibrium (that is, the pressure differencebetween the reservoirs becomes zero). This process is referred as thefirst-stage process 250.

If a rock formation is fractured, its permeability can be characterizedby fracture permeability, in addition to matrix permeability. Suchfractured core samples are described as dual-permeability systemsbecause the system includes two continua and connected transportstructures with different permeability values. FIG. 2C shows adual-permeability system of matrix 201B and fracture 301, and FIG. 2Dshows the corresponding pressure transient curves. For the graph shownin FIG. 2D, the x-axis is time in s, and the y-axis is pressure in psi.As shown in FIG. 2D, the pressure transient curves of a dualpermeability system have a second decline 209 after the first-stageprocess 250. The first-stage process 250 is generally attributed to thecore fracture 301. The second decline 209 is attributed to the smallerpermeability of the core matrix 201B and is referred as the second-stageprocess 270.

Apparatus leakage is a potential source of error in measuringpermeability, and any leakage can have a significant impact on themeasurement, especially for tight rock formations like shales which havevery low permeability. Therefore, it can be useful to correctpermeability assessments of low permeability samples by taking intoaccount any fluid leakage in the measurement system.

FIG. 3A shows the gas flow for a fractured core sample 207B during thefirst-stage process 250 with gas leakage. After the pressure source 109provides the pulse pressure to the upstream reservoir 101, the gas flowsfrom the upstream reservoir 101 into the core fracture 301 and then intothe downstream reservoir 103. During this first-stage process 250, thegas also penetrates the matrix pores of the core sample 207B. Becausethe fracture permeability is much larger than the matrix permeability(usually by orders of magnitude), the impact of the matrix permeabilityis negligible during the first-stage process 250. Leakage rate isdefined as the rate of pressure decrease with time due to leakage inpsi/s, and the leakage rates are identified as α 303 and β 305 for theupstream reservoir 101 and the downstream reservoir 103, respectively.

In the first-stage process 250 of a PDP experiment, the upstream anddownstream pressures are recorded simultaneously. The transient pressureof the system can be modelled mathematically, and the analyticalsolution for the model can be used to determine core samplecharacteristics. Using Darcy's equation and the mass conservationequation, the pressure inside the sample, P(x,t), as a function of thedistance x along the sample and time t can be solved using the followingdifferential equation:

$\begin{matrix}{{\frac{\partial^{2}{P\left( {x,t} \right)}}{\partial x^{2}} = {\frac{c\; \mu \; \varphi_{f}}{k_{f}}\frac{\partial{P\left( {x,t} \right)}}{\partial t}}},{0 < x < L},{t > 0}} & (1) \\{{{P\left( {x,0} \right)} = {P_{2}(0)}},{0 < x < L},} & (2) \\{{{P\left( {0,t} \right)} = {P_{1}(t)}},{t \geq 0},} & (3) \\{{{P\left( {L,t} \right)} = {P_{2}(t)}},{t \geq 0},} & (4)\end{matrix}$

where P₁, P₂, k_(f), c, μ, ϕ_(f), and L are the upstream reservoir 101pressure, the downstream reservoir 103 pressure, the fracturepermeability, the gas compressibility, the gas viscosity, the fractureporosity, and the length of the core sample 107, respectively.

Gas leakage may exist in the testing apparatus, and the massconservation equation at the upper and bottom faces of the core sample107 are:

$\begin{matrix}{{\frac{\partial P_{1}}{\partial t} = {{\frac{k_{f}}{c\; \mu \; \varphi_{f}L}\frac{V_{f}}{V_{1}}\frac{\partial{P\left( {x,t} \right)}}{\partial x}} + \alpha}},{t > 0},{x = 0},} & (5) \\{{\frac{\partial P_{2}}{\partial t} = {{{- \frac{k_{f}}{c\; \mu \; \varphi_{f}L}}\frac{V_{f}}{V_{1}}\frac{\partial{P\left( {x,t} \right)}}{\partial x}} + \beta}},{t > 0},{x = L},} & (6)\end{matrix}$

where V₁, V₂, V_(f), α, and β are the upstream reservoir volume, thedownstream reservoir volume, the fracture volume, the upstream leakagerate, and the downstream leakage rate, respectively.

Some dimensionless parameters are defined to simplify the equations.Dimensionless time (t_(D)) and distance (x_(D)) are defined by Eq. 7.

$\begin{matrix}{{t_{D} = {\frac{k_{f}}{c\; \mu \; \varphi_{f}L^{2}}t}},{x_{D} = \frac{x}{L}}} & (7)\end{matrix}$

The dimensionless parameters a and b are defined by Eq. 8.

$\begin{matrix}{{a = \frac{V_{f}}{V_{1}}},{b = \frac{V_{f}}{V_{2}}}} & (8)\end{matrix}$

The dimensionless gas leakage rates of upstream reservoir 101 anddownstream reservoir 103 are defined by Eqs. 9-10.

$\begin{matrix}{{\alpha_{D\; 0} = {\frac{c\; \mu \; \varphi_{f}L^{2}}{k_{f}}\frac{1}{{P_{1}(0)} - {P_{2}(0)}}\alpha}},} & (9) \\{\beta_{D\; 0} = {\frac{c\; \mu \; \varphi_{f}L^{2}}{k_{f}}\frac{1}{{P_{1}(0)} - {P_{2}(0)}}\beta}} & (10)\end{matrix}$

The normalized pressures of the core sample 107 (P_(D)), the upstreamreservoir 101 (P_(D1)), and the downstream reservoir 103 (P_(D2)) are:

$\begin{matrix}{{{P_{D}\left( {x_{D},t_{D}} \right)} = \frac{{P\left( {x,t} \right)} - {P_{2}(0)}}{{P_{1}(0)} - {P_{2}(0)}}},} & (11) \\{{{P_{D\; 1}\left( t_{D} \right)} = \frac{{P_{1}(t)} - {P_{2}(0)}}{{P_{1}(0)} - {P_{2}(0)}}},} & (12) \\{{{P_{D\; 2}\left( t_{D} \right)} = \frac{{P_{2}(t)} - {P_{2}(0)}}{{P_{1}(0)} - {P_{2}(0)}}},} & (13)\end{matrix}$

where P₂(0) is the initial gas pressure in the PDP system 100, and P₁(0)is the initial gas pressure plus the increased pressure in the upstreamreservoir 101 before gas flow occurs.

With the dimensionless parameters, the differential equations can beconverted to:

$\begin{matrix}{{\frac{\partial^{2}{P_{D}\left( {x_{D},t_{D}} \right)}}{\partial x_{D}^{2}} = \frac{\partial{P_{D}\left( {x_{D},t_{D}} \right)}}{\partial t_{D}}},{0 < x_{D} < 1},{t_{D} > 0},} & (14)\end{matrix}$

with the initial boundary conditions:

$\begin{matrix}{{{P_{D}\left( {x_{D},0} \right)} = 0},{0 < x_{D} < 1},} & (15) \\{{{P_{D}\left( {0,t_{D}} \right)} = {P_{D\; 1}\left( t_{D} \right)}},{t_{D} \geq 0},} & (16) \\{{{P_{D}\left( {1,t_{D}} \right)} = {P_{D\; 2}\left( t_{D} \right)}},{t_{D} \geq 0},} & (17) \\{{\frac{\partial{P_{D\; 1}\left( t_{D} \right)}}{\partial t_{D}} = {{a\frac{\partial{P_{D}\left( {x_{D},t_{D}} \right)}}{\partial x}} + \alpha_{D\; 0}}},{t_{D} \geq 0},{x_{D} = 0},} & (18) \\{{\frac{\partial{P_{D\; 2}\left( t_{D} \right)}}{\partial t_{D}} = {{{- b}\frac{\partial{P_{D}\left( {x_{D},t_{D}} \right)}}{\partial x_{D}}} + \beta_{D\; 0}}},{t_{D} \geq 0},{x_{D} = 1.}} & (19)\end{matrix}$

Through Laplace transform and inverse Laplace transform, the normalizedupstream and downstream pressure with gas leakage are the following:

$\begin{matrix}{{{P_{D\; 1}\left( t_{D} \right)} = {\frac{{P_{1}(t)} - {P_{2}(0)}}{{P_{1}(0)} - {P_{2}(0)}} = {C_{1} + {2{\sum\limits_{n = 1}^{\infty}{e^{({{- t_{D}}\theta_{n}^{2}})}\frac{{\left( {{ab}^{2} + {a\; \theta_{n}^{2}}} \right)\left( {1 - \frac{\alpha_{D\; 0}}{\theta_{n}^{2}}} \right)} + {\frac{{a^{2}b} - {a\; \theta_{n}^{2}}}{\theta_{n}^{2}\cos \; \left( \theta_{n} \right)}\beta_{D\; 0}}}{\theta_{n}^{4} + {\theta_{n}^{2}\left( {a + a^{2} + b + b^{2}} \right)} + {{ab}\left( {a + b + {ab}} \right)}}}}} + {\frac{b\; \alpha_{D\; 0}}{a + b + {ab}}t_{D}} + {\frac{a\; \beta_{D\; 0}}{a + b + {ab}}t_{D}}}}},} & (20) \\{{{P_{D\; 2}\left( t_{D} \right)} = {\frac{{P_{2}(t)} - {P_{2}(0)}}{{P_{1}(0)} - {P_{2}(0)}} = {C_{2} + {2{\sum\limits_{n = 1}^{\infty}{e^{({{- t_{D}}\theta_{n}^{2}})}\frac{{\frac{{ab}^{2} - {b\; \theta_{n}^{2}}}{\cos \left( \theta_{n} \right)}\left( {1 - \frac{\alpha_{D\; 0}}{\theta_{n}^{2}}} \right)} + {\frac{{a^{2}b} + {a\; \theta_{n}^{2}}}{q^{2}}\beta_{D\; 0}}}{\theta_{n}^{4} + {\theta_{n}^{2}\left( {a + a^{2} + b + b^{2}} \right)} + {{ab}\left( {a + b + {ab}} \right)}}}}} + {\frac{b\; \alpha_{D\; 0}}{a + b + {ab}}t_{D}} + {\frac{\alpha \; \beta_{D\; 0}}{a + b + {ab}}t_{D}}}}},} & (21)\end{matrix}$

where C₁ and C₂ are constants calculated by:

$\begin{matrix}{{C_{1} = {1 - {2{\sum\limits_{n = 1}^{\infty}\frac{{\left( {{ab}^{2} + {a\; \theta_{n}^{2}}} \right)\left( {1 - \frac{\alpha_{D\; 0}}{\theta_{n}^{2}}} \right)} + {\frac{{a^{2}b} - {a\; \theta_{n}^{2}}}{\theta_{n}^{2}\cos \; \left( \theta_{n} \right)}\beta_{D\; 0}}}{\theta_{n}^{4} + {\theta_{n}^{2}\left( {a + a^{2} + b + b^{2}} \right)} + {{ab}\left( {a + b + {ab}} \right)}}}}}},} & (22) \\{{C_{2} = {{- 2}{\sum\limits_{n = 1}^{\infty}\frac{{\frac{{ab}^{2} + {b\; \theta_{n}^{2}}}{\cos \left( \theta_{n} \right)}\left( {1 - \frac{\alpha_{D\; 0}}{\theta_{n}^{2}}} \right)} + {\frac{{a^{2}b} + {a\; \theta_{n}^{2}}}{q^{2}}\beta_{D\; 0}}}{\theta_{n}^{4} + {\theta_{n}^{2}\left( {a + a^{2} + b + b^{2}} \right)} + {{ab}\left( {a + b + {ab}} \right)}}}}},} & (23)\end{matrix}$

where θ_(n) are the roots of the following equation:

$\begin{matrix}{{{\tan \left( \theta_{n} \right)} = \frac{\left( {a + b} \right)\theta_{n}}{\theta_{n}^{2} - {ab}}},{\theta_{n} > 0.}} & (24)\end{matrix}$

The leakage rates α 303 or β 305 (or even both) can be zero for no gasleakage conditions and can be attributed to gas desorption, as typicallyassociated with elastomers in the testing apparatus. FIG. 3B is aschematic diagram showing an example of gas flow for a fractured coresample 207B during the first-stage process 250 without gas leakage. Inthis case, any term in the equations above that contains a zero-valuefalls out. For example, if α=0 (no upstream leakage), then α can bedisregarded in Eq. 5, and α_(D0) would equal 0 in Eq. 9, which wouldthen allow for all terms that contain α_(D0) in Eqs. 20-23 to bedisregarded as well. For an PDP experiment without any gas leakage (α=0,β=0), the normalized pressure difference across the core sample 107 iscalculated by the simplified equation:

$\begin{matrix}{{{\ln \left( {\Delta \; P_{D}} \right)} = {{{\ln \left( {P_{D\; 1} - P_{D\; 2}} \right)} \approx {{\ln \left( f_{0} \right)} - {\theta_{1}^{2}t_{D}}}} = {{\ln \left( f_{0} \right)} - {\theta_{1}^{2}\frac{k_{f}}{c\; \mu \; \varphi_{f}L^{2\;}}t}}}},} & (25)\end{matrix}$

where f₀ is a constant. The fracture permeability k_(f) can be estimatedthrough Eq. 25. The same equation can also be used for PDP experimentswith gas leakage, but the estimated permeability would be the apparentfracture permeability k_(af) without correction for gas leakage.

The log of the pressure difference (between the upstream reservoir 101and downstream reservoir 103) is plotted. From the analytical solution,if the measurement system has no gas leakage, this curve is a straightline (see Eq. 25). The slope of this linear line can be used to estimatethe matrix permeability. If there exists gas leakage in the system,however, the curve will bend down with the time measurement, or it willbend up if gas is desorbing from the instrument surfaces. For practicalimplementation, the first derivative of the log of this curve (that is,its slope) is calculated and plotted against time. A derivative linediverging from a constant-derivative line with time (for example, astraight line) can indicate the existence of gas leakage.

If gas leakage exists in the PDP experiment, the permeability estimatescan be corrected to account for the leakage. By solving the physicalmodel with gas leakage, the theoretical results for pressure differencebetween the upstream reservoir 101 and downstream reservoir 103 whichare dependent on the gas leakage rates of the system and thepermeability of the core sample 207A can be obtained. By matching thetheoretical results for the pressure difference as a function of timewith the experimental data, the permeability and the gas leakage ratescan be determined simultaneously.

For instance, a permeability range (k₁, k₂) and gas leakage rate range(α₁, α₂) can be assigned. The analytical solutions for differentpermeability and gas leakage rate combinations within these ranges canbe calculated, along with the squared difference between pressures fromthe analytical solution and the experimental data. The combination ofpermeability k and leakage rate a that provides the least squareddifference, is the corrected estimate of the permeability and gasleakage rate, respectively. The pre-assigned gas leakage rate range canbe determined from knowledge of the system.

The above analytical models can also be applied to the PDP experiment onan unfractured core sample 207A with single matrix permeability becausethe first-stage process 250 is substantially the same. The difference isthat during the application, the fracture permeability k_(f) is replacedwith the matrix permeability k_(m), and all the fracture-relatedparameters are replaced with matrix-related parameters. For instance,the dimensionless parameters a and b in Eq. 8 are converted to:

$\begin{matrix}{{a = \frac{V_{p}}{V_{1\;}}},{b = \frac{V_{p}}{V_{2}}},} & (26)\end{matrix}$

where V_(p) is pore volume of the unfractured core sample 207A. Thesimplification due to α or β (or both) being equal to zero is also thesame for the first-stage process 250 of an unfractured core sample 207A.

After the first-stage process 250 is complete, the experiment for anunfractured core sample 207A is complete. For fractured core samples207B, however, the first-stage process can provide the sample's fracturepermeability, but further analysis of the second-stage process 270 canbe completed to determine the sample's matrix permeability. Referringback to FIG. 2D, after the first-stage process 250 of a PDP experimenton a fracture core sample 207B, the gas flows from the void space (thesum of fracture, upstream reservoir 101, and downstream reservoir 103)into the matrix pores of the core sample 207B, resulting in a secondpressure decline 209. This decay process is defined as the second-stageprocess 270. In the second-stage process 270, the upstream anddownstream pressures are essentially equal. The pressure decline of thewhole system can be modelled mathematically, and the analytical solutionfor the model can be used to determine core sample characteristics.

In order to establish a physical model to describe this second-stageprocess 270, the core sample 207B is approximated as a rectangular cubeinstead of a cylinder. FIG. 4A shows a two-dimensional gas flow model(along the x and z directions), and FIG. 4B shows a simplifiedone-dimensional gas flow model (along the z direction) for thesecond-stage process 270 with leakage rate a 303. In order to obtain theanalytical solutions, some equivalent parameters are defined. The sum ofupstream, downstream and fracture volume can be transformed to anequivalent volume V_(e) 401 as:

V _(e) =V ₁ +V ₂ +V _(f)  (27)

The equivalent matrix height and matrix length are written as

$\begin{matrix}{{h_{me} = \frac{1}{\frac{1}{h_{m\;}} + \frac{1}{L}}},} & (28)\end{matrix}$L _(e) =h _(m) +L,  (29)

where h_(m) is the rock matrix thickness. Since the upstream reservoir101 and downstream reservoir 103 are connected through the fracture, weonly need to assume one leakage rate, a 303.

Using Darcy's equation and the mass conservation equation, thedifferential equation for the pressure inside the core sample 207B,P(z,t), as a function of the distance z across the sample thickness andtime t is calculated by:

$\begin{matrix}{{\frac{\partial^{2}{P\left( {z,t} \right)}}{\partial z^{2\;}} = {\frac{c\; \mu \; \varphi_{m}}{k_{m}}\frac{\partial{P\left( {z,t} \right)}}{\partial t}}},{0 < z < h},{t > t_{1}},{h = \frac{h_{me}}{2}},} & (30)\end{matrix}$

with the initial and boundary conditions:

$\begin{matrix}{{{P\left( {z,t_{1}} \right)} = {P_{0}\left( t_{1} \right)}},{0 < z < h},} & (31) \\{{{P\left( {z,t_{1}} \right)} = {P_{1}\left( t_{1} \right)}},{z = h},} & (32) \\{{\frac{\partial{P\left( {z,t} \right)}}{\partial t} = {{{- \frac{k_{m}}{c\; \mu}}\frac{2d\; L}{V_{e}}\frac{\partial{P\left( {z,t} \right)}}{\partial z}} - \alpha}},{t > t_{1}},{z = h},} & (33) \\{{\frac{\partial{P\left( {z,t} \right)}}{\partial t} = 0},{t > t_{1}},{z = 0},} & (34)\end{matrix}$

where t₁ the first-stage convergence time with gas leakage when theupstream pressure equals to the downstream pressure, and ϕ_(m) is thematrix porosity, excluding the fracture porosity, expressed as:

$\begin{matrix}{\varphi_{m} = {\frac{V_{P}}{V_{b} - V_{f}}.}} & (35)\end{matrix}$

The dimensionless gas leakage rate α_(D1) and dimensionless time τ_(D1)are defined as:

$\begin{matrix}{{\alpha_{D\; 1} = {\frac{c\; \mu \; \varphi_{m}h_{me}^{2}}{4k_{m}}\frac{1}{{P_{1}\left( t_{1} \right)} - {P_{2}(0)}}\alpha}},{\tau_{D\; 1} = {\frac{k_{m}}{c\; \mu \; \varphi_{m}h^{2}}\left( {t - t_{1}} \right)}},} & (36)\end{matrix}$

and ω is defined as:

$\begin{matrix}{{\omega = {\frac{V_{P}}{V_{e}} = \frac{2\; d\; L\; \varphi_{m}h}{V_{e}}}},} & (37)\end{matrix}$

and V_(P) is the volume of the matrix pores.

Through the Laplace transform and inverse Laplace transform, the exactsolution for the normalized pressure of the upstream reservoir 101 anddownstream reservoir 103 for the second-stage process 270 are given by:

$\begin{matrix}{{{U_{D\; 1}\left( \tau_{D\; 1} \right)} = {{U_{D\; 2}\left( \tau_{D\; 1} \right)} = {\frac{{P_{1}(t)} - {P_{0}\left( t_{1} \right)}}{{P_{1}\left( t_{1} \right)} - {P_{0}\left( t_{1} \right)}} = {C_{3} + {2{\sum\limits_{n = 1}^{\infty}{e^{{- \tau_{D\; 1}}\phi_{n}^{2}}\left( {\frac{1}{\omega + \frac{\phi_{n}}{\omega} + 1} - \frac{\omega \; \alpha_{D\; 1}}{\phi_{n}^{4} + \phi_{n}^{2} + {\omega \; \phi_{n}^{2}}}} \right)}}} + \frac{\alpha_{D\; 1} \times \tau_{D\; 1}}{1 + \omega}}}}},} & (38)\end{matrix}$

where C₃ is a constant value which can be calculated by:

$\begin{matrix}{{C_{3} = {1 - {2{\sum\limits_{n = 1}^{\infty}\left( {\frac{1}{\omega + \frac{\phi_{n}}{\omega} + 1} - \frac{\omega \; \alpha_{D\; 1}}{\phi_{n}^{4} + \phi_{n}^{2} + {\omega \; \phi_{n}^{2}}}} \right)}}}},} & (39)\end{matrix}$

and φ_(n) are the roots of the following equation:

$\begin{matrix}{{{\tan \left( \phi_{n} \right)} = {- \frac{\phi_{n}}{\omega}}},{\phi_{n} > 0},} & (40)\end{matrix}$

Similar to the first-stage process, the leakage rate α 303 in thesecond-stage process can be zero for no gas leakage conditions. In thiscase, any term in the equations above that contains a zero-value ratefalls out. In other words, in the case with no leakage, α can bedisregarded in Eq. 33, and α_(D1) would equal 0 in Eq. 36, which wouldthen allow for all terms that contain α_(D1) in Eqs. 38 and 39 to bedisregarded as well. For the second-stage process 270 of a PDPexperiment without any gas leakage (α=0), the matrix permeability k_(m)can be estimated by the following equation:

$\begin{matrix}{{{{\ln \left( {U_{D\; 1} - \frac{1}{1 + \omega}} \right)} \approx {{\ln \left( f_{1} \right)} - {\phi_{1}^{2}\tau_{D\; 1}}}} = {{\ln \left( f_{1} \right)} - {\phi_{1}^{2}\frac{k_{m}}{c\; \mu \; \varphi_{m}L^{2}}\left( {t - t_{1}} \right)}}},} & (41)\end{matrix}$

where f₁ is a constant. The same equation can also be used for PDPexperiments with gas leakage, but the estimated permeability would bethe apparent matrix permeability k_(am) without correction for gasleakage.

In order to detect whether gas leakage exists during the second-stageprocess 270, the log curve of the normalized transient pressure curve isplotted for the second-stage process 270. If there is no gas leakagewith increasing time, this log curve is convergent to a straight line(see Eq. 41), and thus the slope of this line can be used to estimatethe matrix permeability. If there exists gas leakage with increasingtime, however, the log curve will bend down. For practicalimplementation, the first derivative of this log curve (that is, itsslope) is plotted against time. A derivative line diverging from aconstant-derivative line with time (for example, a straight line) canindicate the existence of gas leakage. Similar to analysis of thefirst-stage process, if gas leakage exists in the second-stage processof PDP experiment, the permeability estimate can be corrected to accountfor the leakage by matching the theoretical pressure decline with theexperimental data.

The following transformations can be further applied to the equationsabove to find exact solutions. The Laplace transform is applied on P_(D)in Eq. 42.

(x _(D) ,s)=∫₀ ^(∞) e ^(−s) ² ^(t) ^(D) P _(D)(x _(D) ,t _(D))dt_(D)  (42)

By applying this transform, Eqs. 14, 18, and 19 can be converted to Eqs.43, 44, and 45, respectively.

$\begin{matrix}{{\frac{{\partial^{2}}\left( {x_{D},t_{D}} \right)}{\partial x_{D}^{2}} = {s^{2}\left( {x_{D},t_{D}} \right)}},{0 < x_{D} < 1},{t_{D} > 0},} & (43) \\{{{a\; \frac{\partial \left( {x_{D},t_{D}} \right)}{\partial x_{D}}} = {{s^{2}\left( {x_{D},t_{D}} \right)} - 1 - \frac{\alpha_{D\; 0}}{s^{2}}}},{x_{D} = 0},} & (44) \\{{{b\frac{\partial \left( {x_{D},t_{D}} \right)}{\partial x_{D}}} = {{{- s^{2}}\left( {x_{D},t_{D}} \right)} + \frac{\beta_{D\; 0}}{s^{2}}}},{x_{D} = 1},} & (45)\end{matrix}$

Eq. 43 has the general solution:

(x _(D) ,t _(D))=A sin h(sx _(D))+B cos h(sx _(D)),  (46)

where A and B are constant values that can be determined by the boundaryconditions Eqs. 44 and 45.

By substituting Eq. 46 into Eqs. 44 and 45, A and B can be obtained byEqs. 47 and 48, respectively.

$\begin{matrix}{{A = \frac{{{- {b\left( {s^{2} + \alpha_{D\; 0}} \right)}}{\sinh (s)}} - {{s\left( {s^{2} + \alpha_{D\; 0}} \right)}{\cosh (s)}}}{s^{2}\left( {{{bs}^{2}{\cosh (s)}} + {{abs}\; {\sinh (s)}} + {s^{3}{\sinh (s)}} + {{as}^{2}{\cosh (s)}}} \right)}},} & (47) \\{B = {\frac{{\left( {{bs}^{2} + {\alpha_{D\; 0}b}} \right){\cosh (s)}} + {{s\left( {s^{2} + \alpha_{D\; 0}} \right)}{\sinh (s)}} + {\alpha \; \beta_{D\; 0}}}{s^{2}\left( {{{bs}^{2}{\cosh (s)}} + {{abs}\; {\sinh (s)}} + {s^{3}{\sinh (s)}} + {{as}^{2}{\cosh (s)}}} \right)}.}} & (48)\end{matrix}$

Through the inverse Laplace transform, the dimensionless pressure P_(D)is shown in Eq. 49.

$\begin{matrix}{{P_{D}\left( {x_{D},t_{D}} \right)} = {\frac{1}{2\pi \; i}{\int_{{- \infty}\; i}^{{+ \infty}\; i}{e^{s^{2}t_{D}}\left( {x_{D},s} \right){ds}^{2}}}}} & (49)\end{matrix}$

The complex integral, Eq. 49, can be solved by the Residue Theorem.Thus, the exact solutions for the dimensionless pressure in the upstreamand downstream reservoirs can be expressed as shown in Eqs. 20-24 forthe first-stage process of the PDP experiment. The same method can beapplied to solve the differential equations, Eqs. 30-34, which are usedto describe the second-stage process. The exact solution for thedimensionless pressure in the system can be expressed as shown in Eqs.38-40 for the second-stage process of the PDP experiment.

FIG. 5 shows a flowchart of an example method 500 for performing a PDPexperiment on a core sample retrieved from a formation, according to animplementation. For clarity of presentation, the description thatfollows generally describes method 500 in the context of the otherfigures in this description. At 502, the core sample is prepared for thePDP experiment. For example, the preparation can comprise placing thecore sample in the core holder of the testing apparatus. The preparationcan further comprise supplying test gas to the upstream reservoirutilizing a pressure source, so that the initial pressure is high enoughto diminish the Klinkenberg slipping effect—for example, 1,000 psig.

Once the upstream reservoir and downstream reservoir pressure hasequalized, at 503, a pulse pressure is provided by the pressure sourceto initiate the PDP experiment. From 503, method 500 proceeds to 504,and transient pressure data of the upstream and downstream reservoirsare recorded over time through the first stage as fluid flows from theupstream reservoir to the downstream reservoir. At 506, if the sample isfractured, method 500 proceeds to 508, and transient pressure data ofthe system is recorded over time through the second stage. Method 500then proceeds to 510. If the sample is unfractured, method 500 insteadproceeds directly from 506 to 510, and the pressure data from thefirst-stage process is transformed to detect the presence or absence ofgas leakage in the system. The transformation can comprise normalizingthe pressure of the system, normalizing the pressure difference betweenthe upstream and downstream reservoirs, calculating the logarithm of thenormalized pressure curve, calculating the logarithm of the normalizedpressure difference curve, or a combination of these. The presence of aleak can be determined by detection of a non-straight curve or deviationfrom a substantially straight line. The absence of a leak can bedetermined by detection of a straight (constant-slope) line.

If leakage is detected at 512, then the leakage rate and permeability isestimated at 516 by determining the combination of values that providethe best fit for the analytical model to the experimental data. If noleakage is detected at 512, then the permeability is estimated at 514 byapplying the analytical model. If there was another stage in the PDPexperiment (that is, second-stage process), then method 500 loops backto 510 for the additional stage. The permeability estimates obtainedutilizing method 500 are corrected for any leakage that may exist in thetesting apparatus.

Example 1

A PDP experiment of an implementation is simulated for a hypotheticalunfractured core sample with various upstream leakage rate a. For anunfractured core sample, the PDP experiment comprises the first-stageprocess and determination of the sample's matrix permeability. Theparameters of the unfractured core sample and the PDP system used in thesimulation are provided in Table 1.

Unfractured Tight Core Sample and PDP Experiment Setup

TABLE 1 Parameters Value Unit Core length   1.5 inch Core width 1 inchMatrix porosity 7% Matrix permeability  10⁻⁵ millidarcy (mD) Upstreamvolume (first-stage) 1 cubic centimeter (cc) Downstream volume(first-stage) 1 cc Initial pressure 1,000    psi Pulse pressure 40  psiGas Helium

Gas leakage rates of 0 psi/s, −5×10⁻⁴ psi/s, −5×10⁻³ psi/s, and −1×10⁻²psi/s were simulated for the same unfractured core sample. As shown inFIG. 6A, the leakage rate α affects the pressure transient curve of theupstream reservoir 101 and downstream reservoir 103. Increasing leakagerate can result in shorter convergence time, as well as a decrease inequilibrium pressure. FIG. 6B shows the logarithm of the normalizedpressure difference (ΔP_(D)) curve against time at various gas leakagerates, and FIG. 6C shows the first derivative (signified by anapostrophe) curves of the curves shown on FIG. 6B. In the case of no gasleakage (α=0 psi/s), the logarithm of the normalized pressure differencecurve and its first derivative curve are both straight lines. Fornon-zero gas leakage rates, the logarithm of the normalized pressuredifference curve diverges from a linear line, and its first derivativecurve also diverges from a constant value versus time. Eq. 25 is used tocalculate the apparent matrix permeability at various gas leakage rates.FIG. 6B shows that as gas leakage rate increases, the larger the bias inestimating the matrix permeability, which is actually 10⁻⁵ mD as shownin the table.

FIG. 6D shows a comparison of various analytical solutions of differentmatrix permeability and gas leakage rate combinations against thesimulated measurement data. The logarithm of the normalized pressuredifference (ΔP_(D)) experimental data is plotted. Then, the analyticalsolutions for the log of the normalized pressure difference with variouscombinations of matrix permeability and gas leakage rate are calculated,and the analytical solution that best fits the experimental dataprovides the most accurate estimate of the sample's actual matrixpermeability and the leakage rate of the system.

Example 2

A PDP experiment of an implementation is simulated for a hypotheticalfractured core sample with various upstream leakage rate α. For afractured core sample, the PDP experiment comprises a first-stageprocess for determination of the sample's fracture permeability and asecond-stage process for determination of the sample's matrixpermeability. The physical model and leakage correction method for thefirst-stage process of a fractured sample are similar to those of anunfractured core sample (example above), but the permeability estimatedis attributed to the fracture of the sample, instead of the matrix. Thesecond-stage process of a fractured sample is then explored and analyzedto determine the impact of gas leakage on the estimation of matrixpermeability. During the second-stage process, the pressures within theupstream reservoir, downstream reservoir, and the fracture of the sampleare uniform. The parameters of the fractured core sample and the PDPsystem used in the simulation are provided in Table 2.

Fractured Tight Core Sample and PDP Experiment Setup

TABLE 2 Parameters Value Unit Core length 2 inch Core width 1 inchFracture porosity 0.2% Matrix porosity   7% Fracture permeability   0.1mD Matrix permeability  10⁻⁵ mD Upstream/downstream volume (first-stage)30  cc Upstream/downstream volume (second-stage) 1 cc Initial pressure1,000    psi Pulse pressure 40  psi Gas Helium

After finishing the first-stage process, the volume of the upstream anddownstream reservoirs is reduced to 1 cc to enhance the pressure decaysignal. Gas leakage rates of 0 psi/s, −5×10⁻⁴ psi/s, −5×10⁻³ psi/s, and−1×10⁻² psi/s were simulated for the same fractured core sample. Asshown in FIG. 7A, the leakage rate a affects the pressure transientcurve: increasing leakage rate can result in faster pressure decay rate.FIG. 7B shows the logarithm of the normalized pressure transient curveagainst time at various gas leakage rates (refer to Eq. 41), and FIG. 7Cshows the first derivative curves of the curves shown on FIG. 7B. In thecase of no gas leakage (α=0 psi/s), the logarithm of the normalizedpressure transient curve and its first derivative curve are bothstraight lines. For non-zero gas leakage rates, the logarithm of thenormalized pressure transient curve diverges from a linear line, and itsfirst derivative curve also diverges from a constant value versus time.Eq. 41 is used to calculate the apparent matrix permeability at variousgas leakage rates. FIG. 7B shows that as gas leakage rate increases, thelarger the bias in estimating the matrix permeability, which is actually10⁻⁵ mD as shown in the table.

FIG. 7D shows a comparison of various analytical solutions of differentmatrix permeability and gas leakage rate combinations against thesimulated measurement data. The logarithm of the normalized pressureexperimental data is plotted. Then, the analytical solutions for the logof the normalized pressure with various combinations of matrixpermeability and gas leakage rate are calculated, and the analyticalsolution that best fits the experimental data provides the most accurateestimate of the sample's actual matrix permeability and the leakage rateof the system.

Thus, certain implementations of the subject matter have been described.Other implementations are within the scope of the following claims.

What is claimed is:
 1. A method comprising: performing a pulse-decaypermeability (PDP) experiment on a core sample retrieved from aformation, the PDP experiment comprising flowing fluid through the coresample in a sealed enclosure; measuring, in response to flowing thefluid through the core sample, a change in fluid pressure over time;determining, based on the change in fluid pressure over time, a leakageof fluid from the sealed enclosure; and in response to determining theleakage of fluid from the sealed enclosure, determining an analyticalmodel of the leakage based on the change in fluid pressure over time. 2.The method of claim 1, further comprising determining, based on thechange in fluid pressure over time and on the analytical model of theleakage, a permeability model representing a permeability of the coresample.
 3. The method of claim 2, wherein the permeability is determinedby fitting the non-straight curve with consideration of the leakageeffect.
 4. The method of claim 1, wherein the sealed enclosure comprisesan upstream reservoir, a downstream reservoir, and a core holder betweenthe upstream reservoir and the downstream reservoir, wherein performingthe PDP experiment on the core sample comprises: positioning the coresample in the core holder; and flowing the fluid into the upstreamreservoir, through the core sample in the core holder, and into thedownstream reservoir, wherein the leakage of fluid from the upstream anddownstream reservoirs is determined based on pressure difference betweenan upstream reservoir and a downstream reservoir.
 5. The method of claim4, wherein the core sample is an unfractured core sample.
 6. The methodof claim 5, wherein measuring, in response to flowing the fluid throughthe unfractured core sample, the change in fluid pressure over timecomprises recording pressure transient curves for each of the upstreamand downstream reservoirs.
 7. The method of claim 6, further comprisingdetermining a log of an experimental pressure difference between theupstream and downstream reservoirs based on the pressure transientcurves.
 8. The method of claim 7, wherein determining the leakage of thefluid based on the change in fluid pressure over time comprises:determining that the log of the experimental pressure difference issubstantially a straight line; and determining an absence of the leakagefrom the upstream and downstream reservoirs.
 9. The method of claim 7,wherein determining the leakage of the fluid based on the change influid pressure over time comprises: determining that the log of theexperimental pressure difference is substantially a non-straight curve;and determining presence of the leakage from the upstream and downstreamreservoirs.
 10. The method of claim 9, wherein, in response todetermining the leakage of fluid from the upstream and downstreamreservoirs, determining the leakage rate based on the change in fluidpressure over time by: determining a theoretical pressure differencebetween the upstream and downstream reservoirs based on parameters ofthe fluid flowed through the unfractured core sample, the theoreticalpressure difference being independent of the leakage from the sealedenclosure; and comparing the theoretical pressure difference and theexperimental pressure difference.
 11. The method of claim 4, wherein thecore sample is a fractured core sample comprising a fracture formed in amatrix of the core sample.
 12. The method of claim 11, whereinmeasuring, in response to flowing the fluid through the fractured coresample, the change in fluid pressure over time comprises: measuring afirst-stage change in fluid pressure over time, the first-stage changein fluid pressure based on flow of the fluid through the fracture;measuring a second-stage change in fluid pressure over time, thesecond-stage change in fluid pressure based on flow of the fluid throughthe matrix after the flow of the fluid through the fracture, and whereindetermining, based on the change in fluid pressure over time, theleakage of fluid from the sealed enclosure comprises determining that alog of the second-stage change in fluid pressure over time deviates froma substantially straight line.